1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_2A/multiple/fqus_alt.ma".
16 include "basic_2A/computation/scpds_lift.ma".
17 include "basic_2A/dynamic/snv.ma".
19 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
21 (* Relocation properties ****************************************************)
23 lemma snv_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, g] → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
24 ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
25 #h #g #G #K #T #H elim H -G -K -T
26 [ #G #K #k #L #s #l #m #_ #X #H
27 >(lift_inv_sort1 … H) -X -K -l -m //
28 | #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #l #m #HLK #X #H
29 elim (lift_inv_lref1 … H) * #Hil #H destruct
30 [ elim (drop_trans_le … HLK … HK0) -K /2 width=2 by lt_to_le/ #X #HL0 #H
31 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #L0 #W #HLK0 #HVW #H destruct
32 /3 width=9 by snv_lref/
33 | lapply (drop_trans_ge … HLK … HK0 ?) -K
34 /3 width=9 by snv_lref, drop_inv_gen/
36 | #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #l #m #HLK #X #H
37 elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
38 /4 width=5 by snv_bind, drop_skip/
39 | #a #G #K #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
40 elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
41 elim (lift_total W0 l m)
42 elim (lift_total U0 (l+1) m)
43 /4 width=17 by snv_appl, scpds_lift, lift_bind/
44 | #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
45 elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
46 elim (lift_total U0 l m)
47 /3 width=12 by snv_cast, scpds_lift/
51 lemma snv_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
52 ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ T ¡[h, g].
53 #h #g #G #L #U #H elim H -G -L -U
54 [ #G #L #k #K #s #l #m #_ #X #H
55 >(lift_inv_sort2 … H) -X -L -l -m //
56 | #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #l #m #HLK #X #H
57 elim (lift_inv_lref2 … H) * #Hil #H destruct
58 [ elim (drop_conf_le … HLK … HL0) -L /2 width=2 by lt_to_le/ #X #HK0 #H
59 elim (drop_inv_skip1 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K0 #V #HLK0 #HVW #H destruct
60 /3 width=12 by snv_lref/
61 | lapply (drop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
63 | #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #l #m #HLK #X #H
64 elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
65 /4 width=5 by snv_bind, drop_skip/
66 | #a #G #L #W #W1 #U #U1 #d #_ #_ #HW1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
67 elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
68 elim (scpds_inv_lift1 … HW1 … HLK … HVW) -HW1 #W0 #HW01 #HVW0
69 elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU0
70 elim (lift_inv_bind2 … H) -H #Y #U0 #HY #HU01 #H destruct
71 lapply (lift_inj … HY … HW01) -HY #H destruct
72 /3 width=6 by snv_appl/
73 | #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
74 elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
75 elim (scpds_inv_lift1 … HWU1 … HLK … HVW) -HWU1 #U0 #HU01 #HVU0
76 elim (scpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #HX #HTU0
77 lapply (lift_inj … HX … HU01) -HX #H destruct
78 /3 width=5 by snv_cast/
82 (* Properties on subclosure *************************************************)
84 lemma snv_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
85 ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
86 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
88 elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
89 lapply (drop_inv_O2 … H) -H #H destruct //
91 |5,6: /3 width=8 by snv_inv_lift/
93 [1,3: #a #I #G1 #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
94 |2,4: * #G1 #L1 #V1 #T1 #H
95 [1,3: elim (snv_inv_appl … H) -H //
96 |2,4: elim (snv_inv_cast … H) -H //
101 lemma snv_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
102 ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
103 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_inv_gen … H) -H [|*]
104 /2 width=5 by snv_fqu_conf/
107 lemma snv_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
108 ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
109 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
110 /3 width=5 by fqup_strap1, snv_fqu_conf/
113 lemma snv_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
114 ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
115 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_gen … H) -H [|*]
116 /2 width=5 by snv_fqup_conf/